Abstract
Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup \(G^{\prime }\). We prove that the exponent and the abelianization of the centralizer of \(G^{\prime }\) in G are determined by the group algebra of G over any field of characteristic p. If, additionally, G is 2-generated then almost all the numerical invariants determining G up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of \(G^{\prime }\) is determined. These claims are known to be false for p = 2.
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References
Bagiński, C.: The isomorphism question for modular group algebras of metacyclic p-groups. Proc. Amer. Math. Soc. 104(1), 39–42 (1988)
Bagiński, C.: On the isomorphism problem for modular group algebras of elementary abelian-by-cyclic p-groups. Colloq. Math. 82(1), 125–136 (1999)
Bagiński, C., Caranti, A.: The modular group algebras of p-groups of maximal class. Canad. J. Math. 40(6), 1422–1435 (1988)
Broche, O., del Río, Á.: The Modular Isomorphism Problem for two generated groups of class two. Indian J. Pure Appl. Math. 52, 721–728 (2021)
Broche, O., García-Lucas, D., del Río, Á.: A classification of the finite two-generated cyclic-by-abelian groups of prime power order, arXiv:2106.06449
Bagiński, C., Konovalov, A.: The modular isomorphism problem for finite p-groups with a cyclic subgroup of index p2, Groups St. Andrews2005. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 339, Cambridge Univ. Press, Cambridge, pp. 186–193 (2007)
Bleher, F.M., Kimmerle, W., Roggenkamp, K.W., Wursthorn, M.: Computational Aspects of the Isomorphism Problem, Algorithmic Algebra and Number Theory (Heidelberg, 1997), pp 313–329. Springer, Berlin (1999)
Brauer, R: Representations of Finite Groups, Lectures on Modern Mathematics, vol. I, pp 133–175. Wiley, N. Y. (1963)
Cheng, Y.: On finite p-groups with cyclic commutator subgroup. Arch. Math. 39(4), 295–298 (1982)
Curtis, C. h. W., Reiner, I.: Methods of Representation Theory, vol. I. Wiley, New York (1981). With applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication. MR 632548 (82i:20001)
Curtis, C.h.W., Reiner, I.: Methods of Representation Theory, vol. II. Pure and Applied Mathematics (New York). Wiley, New York (1987). With applications to finite groups and orders, A Wiley-Interscience Publication, MR 892316 (88f:20002)
Dade, E: Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps. Math. Z. 119, 345–348 (1971)
Deskins, W.E.: Finite Abelian groups with isomorphic group algebras. Duke Math. J. 23, 35–40 (1956). MR 77535
Eick, B.: Computing automorphism groups and testing isomorphisms for modular group algebras. J. Algebra 320(11), 3895–3910 (2008)
Eick, B., Konovalov, A.: The modular isomorphism problem for the groups of order 512, Groups St Andrews 2009 in Bath. Volume 2, London, Math. Soc. Lecture Note Ser., vol. 388, Cambridge Univ. Press, Cambridge, pp. 375–383 (2011)
García-Lucas, D., Margolis, L., del Río, Á.: Non-isomorphic 2-groups with isomorphic modular group algebras. J. Reine Angew. Math. 154(783), 269–274 (2022)
Hertweck, M.: A counterexample to the isomorphism problem for integral group rings. Ann. Math. (2) 154(1), 115–138 (2001)
Higman, G.: The units of group-rings. Proc. London Math. Soc. (2) 46, 231–248 (1940)
Hertweck, M., Soriano, M: On the modular isomorphism problem: Groups of order 26. Groups, rings and algebras. Contemp. Math., vol. 420, pp 177–213. Amer. Math. Soc., Providence (2006)
Hertweck, M, Soriano, M: Parametrization of central Frattini extensions and isomorphisms of small group rings. Israel J. Math. 157, 63–102 (2007)
Huppert, B.: Endliche Gruppen I. Die Grundlehren der Mathematischen Wissenschaften Band, vol. 134. Springer, Berlin (1967)
Jennings, S.A.: The structure of the group ring of a p-group over a modular field. Trans. Amer. Math. Soc. 50, 175–185 (1941)
Külshammer, B.: Bemerkungen über die Gruppenalgebra als symmetrische Algebra II. J. Algebra 75(1), 59–69 (1982)
Kimmerle, W.: Beiträge zur ganzzahligen Darstellungstheorie endlicher Gruppen. Bayreuth Math. Schr. 36, 139 (1991)
Makasikis, A.: Sur l’isomorphie d’algèbres de groupes sur un champ modulaire. Bull. Soc. Math. Belg. 28(2), 91–109 (1976). MR 561324
Margolis, L: The Modular Isomorphism Problem: A Survey, Jahresber. Dtsch. Math Ver (2022)
Margolis, L, Moede, T.: The Modular Isomorphism Problem for small groups – revisiting Eick’s algorithm. arXiv:https://arxiv.org/abs/2010.07030
Margolis, L., Stanojkovski, M.: On the modular isomorphism problem for groups of class 3 and obelisks. J. Group Theory 25(1), 163–206 (2022)
Margolis, L., Stanojkovski, M., Sakurai, T: Abelian invariants and a reduction theorem for the modular isomorphism problem, arXiv:https://arxiv.org/abs/2110.10025
Navarro, G., Sambale, B.: On the blockwise modular isomorphism problem. Manuscripta Math. 157(1–2), 263–278 (2018)
Passman, D.S.: Isomorphic groups and group rings. Pacific J. Math. 15, 561–583 (1965)
Passman, D.S.: The Algebraic Structure of Group Rings, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney (1977)
Perlis, S., Walker, G.L.: Abelian group algebras of finite order. Trans. Amer. Math. Soc. 68, 420–426 (1950)
Quillen, D.G.: On the associated graded ring of a group ring. J. Algebra 10, 411–418 (1968)
Roggenkamp, K.W., Scott, L.: Isomorphisms of p-adic group rings. Ann. of Math. (2) 126(3), 593–647 (1987)
Sakurai, T: The isomorphism problem for group algebras: A criterion. J. Group Theory 23(3), 435–445 (2020)
Salim, A.M.M: The Isomorphism Problem for the Modular Group Algebras of Groups of Order p5. ProQuest LLC, Ann Arbor (1993). Thesis (Ph.D.)–University of Manchester
Sandling, R: Units in the modular group algebra of a finite abelian p-group. J. Pure Appl. Algebra 33(3), 337–346 (1984)
Sandling, R: The Isomorphism Problem for Group Rings: A survey, Orders and their Applications (Oberwolfach, 1984), Lecture Notes in Math., vol. 1142, pp 256–288. Springer, Berlin (1985)
Sandling, R.: The modular group algebra of a central-elementary-by-abelian p-group. Arch. Math. (Basel) 52(1), 22–27 (1989)
Sandling, R.: The modular group algebra problem for metacyclic p-groups. Proc. Amer. Math. Soc. 124(5), 1347–1350 (1996)
Sehgal, S.K.: On the isomorphism of group algebras. Math. Z. 95, 71–75 (1967)
Sehgal, S.K.: Topics in group rings. Monographs and Textbooks in Pure and Applied Math., vol. 50. Marcel Dekker, Inc., New York (1978)
Song, Q: Finite two-generator p-subgrous with cyclic derived group. Comm. Algebra 41(4), 1499–1513 (2013)
Salim, M.A.M., Sandling, R.: The unit group of the modular small group algebra. Math. J. Okayama Univ. 37(1995), 15–25 (1996)
Salim, M.A.M., Sandling, R.: The modular group algebra problem for groups of order p5. J. Austral. Math. Soc. Ser. A 61(2), 229–237 (1996)
Ward, H.N.: Some results on the group algebra of a p-group over a prime field, Seminar on finite groups and related topics., Mimeographed notes, Harvard Univ., pp. 13–19 (1960)
Whitcomb, A.: The Group Ring Problem. Thesis (Ph.D.)–The University of Chicago, Ann Arbor (1968)
Wursthorn, M.: Isomorphisms of modular group algebras: An algorithm and its application to groups of order 26. J. Symbolic Comput. 15(2), 211–227 (1993). MR 1218760
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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The first two authors are partially supported by Grant PID2020-113206GB-I00 funded by MCIN/ AEI/10.13039/501100011033. The third author is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 286237555 – TRR 195.
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García-Lucas, D., del Río, Á. & Stanojkovski, M. On Group Invariants Determined by Modular Group Algebras: Even Versus Odd Characteristic. Algebr Represent Theor 26, 2683–2707 (2023). https://doi.org/10.1007/s10468-022-10182-x
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DOI: https://doi.org/10.1007/s10468-022-10182-x